Question
A rational function has the given information. Choose the equation that could represent the same rational function.
horizontal asymptote: y=32
vertical asymptote: x=−2
hole at x=5
(1 point)
Responses
y=3x2−15x2x2 − 10x
y=3x2−15x2x2 − 10x
y=3x2−12x−152x2−6x−20
y is equal to the fraction with numerator 3 x squared minus 12 x minus 15 and denominator 2 x squared minus 6 x minus 20
y=3x4−12x−152x6−6x3−20x
y is equal to the fraction with numerator 3 x to the 4th power minus 12 x minus 15 and denominator 2 x to the 6th power minus 6 x cubed minus 20 x
y=x2−4x−5x2−3x−10
horizontal asymptote: y=32
vertical asymptote: x=−2
hole at x=5
(1 point)
Responses
y=3x2−15x2x2 − 10x
y=3x2−15x2x2 − 10x
y=3x2−12x−152x2−6x−20
y is equal to the fraction with numerator 3 x squared minus 12 x minus 15 and denominator 2 x squared minus 6 x minus 20
y=3x4−12x−152x6−6x3−20x
y is equal to the fraction with numerator 3 x to the 4th power minus 12 x minus 15 and denominator 2 x to the 6th power minus 6 x cubed minus 20 x
y=x2−4x−5x2−3x−10
Answers
GPT-4o mini
To find a rational function that matches the given properties, we need to pay attention to the requirements provided:
1. **Horizontal Asymptote (y = 32)**: For a rational function, if the degrees of the numerator and denominator are the same, then the horizontal asymptote can be found by taking the ratio of the leading coefficients of the numerator and denominator. Therefore, we want the leading coefficient ratio to equal 32.
2. **Vertical Asymptote (x = -2)**: There should be a factor in the denominator that creates a vertical asymptote. Specifically, the denominator must have a term that equals zero when x = -2, so it should contain the factor (x + 2).
3. **Hole at (x = 5)**: A hole at x = 5 means that both the numerator and denominator of the rational function should have the factor (x - 5), so that the function is undefined at that point, but the graph does not have a vertical asymptote there.
Based on these requirements, let's analyze the options given:
1. **y = (3x² - 15) / (2x² - 10x)**:
- This does not have a hole at x = 5 (numerator and denominator cannot be simplified by (x - 5)).
- Vertical asymptote does not come out right.
2. **y = (3x² - 12x - 15) / (2x² - 6x - 20)**:
- Investigating factors, for the numerator: 3x² - 12x - 15 factors cleanly to (3)(x - 5)(x + 1).
- The denominator: 2x² - 6x - 20 factors to 2(x + 2)(x - 5), exhibiting a hole at x = 5, and thus vertical asymptote at x = -2.
Check Coefficients: Leading coefficients => 3 / 2 (not matching), however, numerator can be adjusted to yield our required horizontal asymptote.
3. **y = (3x^4 - 12x - 15) / (2x^6 - 6x^3 - 20x)**:
- This does not match based on complexity with higher degrees involved, and certainly does not yield direct simplification.
4. **y = (x² - 4x - 5) / (x² - 3x - 10)**:
- Not matching the vertical asymptote or the required hole.
Given this analysis, **the best representation** based on the conditions — with some modifications needed for horizontal asymptote — appears to be:
**y = (3x² - 12x - 15) / (2x² - 6x - 20)**.
Thus, if strictly following all criteria produced here, ensure that alterations are made accurately to lead coefficients to maintain 32 for \(\frac{3}{2}\) placements being strict vertical evaluations.
1. **Horizontal Asymptote (y = 32)**: For a rational function, if the degrees of the numerator and denominator are the same, then the horizontal asymptote can be found by taking the ratio of the leading coefficients of the numerator and denominator. Therefore, we want the leading coefficient ratio to equal 32.
2. **Vertical Asymptote (x = -2)**: There should be a factor in the denominator that creates a vertical asymptote. Specifically, the denominator must have a term that equals zero when x = -2, so it should contain the factor (x + 2).
3. **Hole at (x = 5)**: A hole at x = 5 means that both the numerator and denominator of the rational function should have the factor (x - 5), so that the function is undefined at that point, but the graph does not have a vertical asymptote there.
Based on these requirements, let's analyze the options given:
1. **y = (3x² - 15) / (2x² - 10x)**:
- This does not have a hole at x = 5 (numerator and denominator cannot be simplified by (x - 5)).
- Vertical asymptote does not come out right.
2. **y = (3x² - 12x - 15) / (2x² - 6x - 20)**:
- Investigating factors, for the numerator: 3x² - 12x - 15 factors cleanly to (3)(x - 5)(x + 1).
- The denominator: 2x² - 6x - 20 factors to 2(x + 2)(x - 5), exhibiting a hole at x = 5, and thus vertical asymptote at x = -2.
Check Coefficients: Leading coefficients => 3 / 2 (not matching), however, numerator can be adjusted to yield our required horizontal asymptote.
3. **y = (3x^4 - 12x - 15) / (2x^6 - 6x^3 - 20x)**:
- This does not match based on complexity with higher degrees involved, and certainly does not yield direct simplification.
4. **y = (x² - 4x - 5) / (x² - 3x - 10)**:
- Not matching the vertical asymptote or the required hole.
Given this analysis, **the best representation** based on the conditions — with some modifications needed for horizontal asymptote — appears to be:
**y = (3x² - 12x - 15) / (2x² - 6x - 20)**.
Thus, if strictly following all criteria produced here, ensure that alterations are made accurately to lead coefficients to maintain 32 for \(\frac{3}{2}\) placements being strict vertical evaluations.